I don't know if one can say a priori if a normal distribution is expected. But the fact is many sets of numbers from different fields are normally dististributed, but many aren't.
There are statistical tests for normalcy that you can apply to your data, which I'm not going to do here. I'll just present two plots.
The first is the average daily temperature in my town of Salem, Oregon. I've been collecting it daily since I moved here in September 2012, and was able to easily obtain the numbers for the 12 months prior. Here's a histogram of the distribution of the day's deviation from average (defined over the interval 1981-2010).
The bins are 0.5°F wide, and instead of plotting the count of the number of temperatures in each bin I've plotted the percentage of them amount the total. It looks fairly Gaussian (="normal"), but with the peak count shifted a couple of degrees to the right, which I assume is global warming.
Then I've plotted the monthly anomalies for HadCRUT4 over the length of their record, which starts in January 1850 (6/12 8:30 am PT: this has been corrected since the original last night; the results didn't change sustantially):
These data clearly aren't normally distributed. The long length of the record, most of which is before the sharp change in manmade global warming in 1975 or so, skews the dataset.
It's late and I'll try to write more about this tomorrow. Corrections and comments welcome.